# Taylor series.

Definition 10.5.1   Let be a function having derivatives of any order at . The power series

is called the Taylor series of the function at .

If , this series is called the Mac-Laurin series of .

Example 10.5.2 (A couple of MacLaurin series)

• , for .

Theorem 10.5.3   Let and be two functions having Taylor series expansions and respectively, converging on the interval . Then has as its Taylor series expansion the series and it converges on .

Theorem 10.5.4   Let and be two functions having Taylor series expansions and respectively, converging on the interval . Then has as its Taylor series expansion the product of these series and it converges on .

Example 10.5.5   On the one side we have:

On the other side, we have:

 Thus, we have: